Note that the coupling phase \(\theta \)  of the direct interaction should be considered (i.e., \(J=|J|e^{i\theta }\) ), which can not be removed by any gauge transformation in the presence of the waveguide mediated coupling. As will be shown in the following, it plays a crucial role for achieving nonreciprocal excitation transfer. Experimentally, such a coupling phase can be achieved via an ac driving in each emitter [1]}, [2]}, [3]}, [4]}. For whispering gallery mode resonators, one can also use an anti-resonant linker to introduce an optical path imbalance \(\Delta x\)  in opposite directions between the two resonators, such that the effective coupling phase reads \(\theta =2\pi \Delta x/\lambda \)  with \(\lambda \)  the resonant wavelength of the resonators [5]}, [6]}, [7]}, [8]}.
